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Journal articles on the topic 'Solving polynomial systems of equations'

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Published: 3 June 2025

Last updated: 31 July 2025

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1

Manocha, D. "Solving systems of polynomial equations." IEEE Computer Graphics and Applications 14, no. 2 (1994): 46–55. http://dx.doi.org/10.1109/38.267470.

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2

Moszyński, Krzysztof. "Remarks on polynomial methods for solving systems of linear algebraic equations." Applications of Mathematics 37, no. 6 (1992): 419–36. http://dx.doi.org/10.21136/am.1992.104521.

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3

LLIBRE, JAUME, and CLAUDIA VALLS. "POLYNOMIAL FIRST INTEGRALS FOR THE CHEN AND LÜ SYSTEMS." International Journal of Bifurcation and Chaos 22, no. 11 (2012): 1250262. http://dx.doi.org/10.1142/s0218127412502628.

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We characterize all the values of the parameters for which the Chen and Lü systems have polynomial first integrals by using weight homogeneous polynomials and the method of characteristics for solving partial differential equations. We improve previous results which were not complete.

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4

Wang, Dongming. "Solving polynomial equations: Characteristic sets and triangular systems." Mathematics and Computers in Simulation 42, no. 4-6 (1996): 339–51. http://dx.doi.org/10.1016/s0378-4754(96)00008-0.

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5

Serdyukova, S. I. "Solving large systems of polynomial equations using REDUCE." Programming and Computer Software 26, no. 1 (2000): 28–29. http://dx.doi.org/10.1007/bf02759175.

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6

Chukanov, Sergei Nikolaevich, and Ilya Stanislavovich Chukanov. "The Investigation of Nonlinear Polynomial Control Systems." Modeling and Analysis of Information Systems 28, no. 3 (2021): 238–49. http://dx.doi.org/10.18255/1818-1015-2021-3-238-249.

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The paper considers methods for estimating stability using Lyapunov functions, which are used for nonlinear polynomial control systems. The apparatus of the Gro¨bner basis method is used to assess the stability of a dynamical system. A description of the Gro¨bner basis method is given. To apply the method, the canonical relations of the nonlinear system are approximated by polynomials of the components of the state and control vectors. To calculate the Gro¨bner basis, the Buchberger algorithm is used, which is implemented in symbolic computation programs for solving systems of nonlinear polyno

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7

Raghavan, M., and B. Roth. "Solving Polynomial Systems for the Kinematic Analysis and Synthesis of Mechanisms and Robot Manipulators." Journal of Mechanical Design 117, B (1995): 71–79. http://dx.doi.org/10.1115/1.2836473.

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Problems in mechanisms analysis and synthesis and robotics lead naturally to systems of polynomial equations. This paper reviews the state of the art in the solution of such systems of equations. Three well-known methods for solving systems of polynomial equations, viz., Dialytic Elimination, Polynomial Continuation, and Grobner bases are reviewed. The methods are illustrated by means of simple examples. We also review important kinematic analysis and synthesis problems and their solutions using these mathematical procedures.

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Raghavan, M., and B. Roth. "Solving Polynomial Systems for the Kinematic Analysis and Synthesis of Mechanisms and Robot Manipulators." Journal of Vibration and Acoustics 117, B (1995): 71–79. http://dx.doi.org/10.1115/1.2838679.

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Problems in mechanisms analysis and synthesis and robotics lead naturally to systems of polynomial equations. This paper reviews the state of the art in the solution of such systems of equations. Three well-known methods for solving systems of polynomial equations, viz., Dialytic Elimination, Polynomial Continuation, and Grobner bases are reviewed. The methods are illustrated by means of simple examples. We also review important kinematic analysis and synthesis problems and their solutions using these mathematical procedures.

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9

Barotov, Dostonjon Numonjonovich, and Ruziboy Numonjonovich Barotov. "Polylinear Transformation Method for Solving Systems of Logical Equations." Mathematics 10, no. 6 (2022): 918. http://dx.doi.org/10.3390/math10060918.

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In connection with applications, the solution of a system of logical equations plays an important role in computational mathematics and in many other areas. As a result, many new directions and algorithms for solving systems of logical equations are being developed. One of these directions is transformation into the real continuous domain. The real continuous domain is a richer domain to work with because it features many algorithms, which are well designed. In this study, firstly, we transformed any system of logical equations in the unit n-dimensional cube Kn into a system of polylinear–poly

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10

Rojas, J. Maurice, and Yuyu Zhu. "A complexity chasm for solving sparse polynomial equations over p -adic fields." ACM Communications in Computer Algebra 54, no. 3 (2020): 86–90. http://dx.doi.org/10.1145/3457341.3457343.

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The applications of solving systems of polynomial equations are legion: The real case permeates all of non-linear optimization as well as numerous problems in engineering. The p -adic case leads to many classical questions in number theory, and is close to many applications in cryptography, coding theory, and computational number theory. As such, it is important to understand the complexity of solving systems of polynomial equations over local fields. Furthermore, the complexity of solving structured systems --- such as those with a fixed number of monomial terms or invariance with respect to

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11

Ivanyos, Gábor, and Miklos Santha. "Solving systems of diagonal polynomial equations over finite fields." Theoretical Computer Science 657 (January 2017): 73–85. http://dx.doi.org/10.1016/j.tcs.2016.04.045.

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12

Farahani, Hamed, and Hossein Jafari. "SOLVING FULLY FUZZY POLYNOMIAL EQUATIONS SYSTEMS USING EIGENVALUE METHOD." Advances in Fuzzy Sets and Systems 24, no. 1 (2019): 29–54. http://dx.doi.org/10.17654/fs024010029.

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13

Barotov, Dostonjon, Aleksey Osipov, Sergey Korchagin, et al. "Transformation Method for Solving System of Boolean Algebraic Equations." Mathematics 9, no. 24 (2021): 3299. http://dx.doi.org/10.3390/math9243299.

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In recent years, various methods and directions for solving a system of Boolean algebraic equations have been invented, and now they are being very actively investigated. One of these directions is the method of transforming a system of Boolean algebraic equations, given over a ring of Boolean polynomials, into systems of equations over a field of real numbers, and various optimization methods can be applied to these systems. In this paper, we propose a new transformation method for Solving Systems of Boolean Algebraic Equations (SBAE). The essence of the proposed method is that firstly, SBAE

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14

Jiang, Zhaolin. "Fast Algorithms for Solving FLSR-Factor Block Circulant Linear Systems and Inverse Problem ofAX=b." Abstract and Applied Analysis 2014 (2014): 1–9. http://dx.doi.org/10.1155/2014/340803.

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Block circulant and circulant matrices have already become an ideal research area for solving various differential equations. In this paper, we give the definition and the basic properties of FLSR-factor block circulant (retrocirculant) matrix over fieldF. Fast algorithms for solving systems of linear equations involving these matrices are presented by the fast algorithm for computing matrix polynomials. The unique solution is obtained when such matrix over a fieldFis nonsingular. Fast algorithms for solving the unique solution of the inverse problem ofAX=bin the class of the level-2 FLS(R,r)-

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15

Alshabanat, Amal, and Bessem Samet. "A numerical study of a coupled system of fractional differential equations." Filomat 34, no. 8 (2020): 2585–600. http://dx.doi.org/10.2298/fil2008585a.

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We consider a certain class of coupled systems of fractional differential equations involving ?-Caputo fractional derivatives. A numerical approach is provided for solving this class of systems. The method is based on operational matrix of fractional integration of an arbitrary ?-polynomial basis. A theoretical study related to the numerical scheme and the convergence of the method is presented. Next, several numerical examples are given using different types of polynomials aiming to confirm the efficiency of our approach.

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16

Nedashkovska, Anastasiya. "Solving systems of matrix equations of the second degree." Physico-mathematical modelling and informational technologies, no. 33 (September 3, 2021): 52–56. http://dx.doi.org/10.15407/fmmit2021.33.052.

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Matrix equations and systems of matrix equations are widely used in control system optimization problems. However, the methods for their solving are developed only for the most popular matrix equations – Riccati and Lyapunov equations, and there is no universal approach for solving problems of this class. This paper summarizes the previously considered method of solving systems of algebraic equations over a field of real numbers [1] and proposes a scheme for systems of polynomial matrix equations of the second degree with many unknowns. A recurrent formula for fractionalization a solution into

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17

Wampler, C. W., A. P. Morgan, and A. J. Sommese. "Numerical Continuation Methods for Solving Polynomial Systems Arising in Kinematics." Journal of Mechanical Design 112, no. 1 (1990): 59–68. http://dx.doi.org/10.1115/1.2912579.

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Many problems in mechanism design and theoretical kinematics can be formulated as systems of polynomial equations. Recent developments in numerical continuation have led to algorithms that compute all solutions to polynomial systems of moderate size. Despite the immediate relevance of these methods, they are unfamiliar to most kinematicians. This paper attempts to bridge that gap by presenting a tutorial on the main ideas of polynomial continuation along with a section surveying advanced techniques. A seven position Burmester problem serves to illustrate the basic material and the inverse posi

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18

Woba, Moumouni Djassibo. "Solving Some Problems and Elimination in Systems of Polynomial Equations." American Journal of Computational Mathematics 14, no. 03 (2024): 333–45. http://dx.doi.org/10.4236/ajcm.2024.143016.

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19

Bates, Daniel J., Andrew J. Newell, and Matthew Niemerg. "BertiniLab: A MATLAB interface for solving systems of polynomial equations." Numerical Algorithms 71, no. 1 (2015): 229–44. http://dx.doi.org/10.1007/s11075-015-0014-6.

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20

Anil Kumar, Sachin Kumar. "New Operational Matrix Via Gnocchi Polynomial for Solving Non-Linear Fractional Differential Equations." Communications on Applied Nonlinear Analysis 32, no. 9s (2025): 2969–81. https://doi.org/10.52783/cana.v32.4595.

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Fractional differential equations (FDEs) have emerged as essential tools in modeling complex dynamical systems exhibiting memory and hereditary properties. Traditional operational matrices arising from Legendre, Chebyshev, and Jacobi polynomials are generally known to be numerically unstable, computationally expensive, and inefficient in approximating fractional operators. In this study an operational matrix based on Gnocchi polynomial is introduced for solving non linear fractional differential equations (NFDE) with better sparsity, stability and computational efficiency. The proposed method

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21

Yannakoudakis, Aristotle G. "Full Static Output Feedback Equivalence." Journal of Control Science and Engineering 2013 (2013): 1–17. http://dx.doi.org/10.1155/2013/491709.

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We present a constructive solution to the problem of full output feedback equivalence, of linear, minimal, time-invariant systems. The equivalence relation on the set of systems is transformed to another on the set of invertible block Bezout/Hankel matrices using the isotropy subgroups of the full state feedback group and the full output injection group. The transformation achieving equivalence is calculated solving linear systems of equations. We give a polynomial version of the results proving that two systems are full output feedback equivalent, if and only if they have the same family of g

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22

Khakimova, Zilya Nailevna, Larisa Nikolaevna Timofeeva, and Ajkanush Ashotovna Atojan. "Applying a Power Transformation to the Orbit of the 2nd Painleve Equation and Solving Differential Equations with Polynomial Right-hand Sides Via the 2nd Painleve Transcendent and in Polynomials." Differential Equations and Control Processes, no. 4 (2023): 142–54. http://dx.doi.org/10.21638/11701/spbu35.2023.407.

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The 2nd Painleve equation is considered as a representative of the second-order class of ordinary differential equations (ODEs) with polynomial right-hand sides, as well as of the more general second-order class of equations with fractional polynomial right-hand sides. The second PainlevГ© equation with three terms on the right side has an orbit in the class of fractional polynomial equations with respect to the pseudogroup of the 36th order, and in the absence of the 3rd term вЂ“ the 60th order. This paper presents a power transformation with an arbitrary parameter that preserves the polynomi

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23

Barotov, Dostonjon N., and Ruziboy N. Barotov. "Construction of smooth convex extensions of Boolean functions." Russian Universities Reports. Mathematics, no. 145 (2024): 20–28. http://dx.doi.org/10.20310/2686-9667-2024-29-145-20-28.

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Systems of Boolean equations are widely used in mathematics, computer science, and applied sciences. In this regard, on the one hand, new research methods and algorithms are being developed for such systems, and on the other hand, existing methods and algorithms for solving such systems are being improved. One of these methods is that, firstly, the system of Boolean equations given over the ring of Boolean polynomials is transformed into a system of equations over the field of real numbers, and secondly, the transformed system is reduced either to the problem of numerical minimization of the c

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24

Zajac, Pavol. "MRHS Equation Systems that can be Solved in Polynomial Time." Tatra Mountains Mathematical Publications 67, no. 1 (2016): 205–19. http://dx.doi.org/10.1515/tmmp-2016-0040.

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Abstract In this article we study the difficulty of solving Multiple Right-Hand Side (MRHS) equation systems. In the first part we show that, in general, solving MRHS systems is NP-hard. In the next part we focus on special (large) families of MRHS systems that can be solved in polynomial time with two algorithms: one based on linearisation of MRHS equations, and the second one based on decoding problems that can be solved in polynomial time.

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25

Il’ev, A. V., and V. P. Il’ev. "ALGORITHMS FOR SOLVING SYSTEMS OF EQUATIONS OVER VARIOUS CLASSES OF FINITE GRAPHS." Prikladnaya Diskretnaya Matematika, no. 53 (2021): 89–102. http://dx.doi.org/10.17223/20710410/53/6.

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The aim of the paper is to study and to solve finite systems of equations over finite undirected graphs. Equations over graphs are atomic formulas of the language L consisting of the set of constants (graph vertices), the binary vertex adjacency predicate and the equality predicate. It is proved that the problem of checking compatibility of a system of equations S with k variables over an arbitrary simple n-vertex graph Γ is N P-complete. The computational complexity of the procedure for checking compatibility of a system of equations S over a simple graph Γ and the procedure for finding a gen

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26

GERDT, VLADIMIR P. "COMPUTER ALGEBRA, SYMMETRY ANALYSIS AND INTEGRABILITY OF NONLINEAR EVOLUTION EQUATIONS." International Journal of Modern Physics C 04, no. 02 (1993): 279–86. http://dx.doi.org/10.1142/s012918319300029x.

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A computer algebra-aided symmetry approach to investigating integrability of polynomial-nonlinear evolution equations in one-temporal and one-spatial dimensions is presented. The approach is based on verifying the existence of higher conservation laws and symmetries. If the equations contain arbitrary numerical parameters, the problem of selection of all the integrable cases is reduced to the solving polynomial equations in those parameters. The Gröbner basis technique is used in order to simplify and to solve such polynomial systems which typically have infinitely many solutions.

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27

Nigim, K. A., M. M. A. Salama, and M. Kazerani. "Solving Polynomial Algebraic Equations of the Stand Alone Induction Generator." International Journal of Electrical Engineering & Education 40, no. 1 (2003): 45–54. http://dx.doi.org/10.7227/ijeee.40.1.5.

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This paper describes the use of MathCAD's solving block which uses ‘Given’ and ‘Find’ built-in functions to solve nth order nonlinear algebraic equations. Introducing complex energy systems to electrical engineering students in their undergraduate studies is essential to complement many energy conversion courses. Various electric energy-capturing schemes use electric equivalent circuit models that incorporate nonlinear elements with complex mathematical formulas requiring numerical computation. Without incorporating programming tools, the taught material could be vague and a burden for both th

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28

Malaschonok, Natasha. "Solving Differential Equations by Parallel Laplace Method with Assured Accuracy." Serdica Journal of Computing 1, no. 4 (2007): 387–402. http://dx.doi.org/10.55630/sjc.2007.1.387-402.

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We produce a parallel algorithm realizing the Laplace transform method for the symbolic solving of differential equations. In this paper we consider systems of ordinary linear differential equations with constant coefficients, nonzero initial conditions and right-hand parts reduced to sums of exponents with polynomial coefficients.

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29

Pichuev, K. D., and A. N. Rybalov. "On the complexity of solving of equations in the bicyclic monoid." Herald of Omsk University 29, no. 1 (2024): 8–17. http://dx.doi.org/10.24147/1812-3996.2024.1.8-17.

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In this article we prove that the problem of solvability of systems of equations over the bicyclic a monoid is NP-hard. On the other hand, we prove polynomial decidability of this problem for some natural class of equations in one variable.

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30

Lichtblau, Daniel. "Approximate Gröbner Bases, Overdetermined Polynomial Systems, and Approximate GCDs." ISRN Computational Mathematics 2013 (March 21, 2013): 1–12. http://dx.doi.org/10.1155/2013/352806.

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We discuss computation of Gröbner bases using approximate arithmetic for coefficients. We show how certain considerations of tolerance, corresponding roughly to absolute and relative error from numeric computation, allow us to obtain good approximate solutions to problems that are overdetermined. We provide examples of solving overdetermined systems of polynomial equations. As a secondary feature we show handling of approximate polynomial GCD computations, using benchmarks from the literature.

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31

Parkinson, Suzanna, Hayden Ringer, Kate Wall, et al. "Analysis of normal-form algorithms for solving systems of polynomial equations." Journal of Computational and Applied Mathematics 411 (September 2022): 114235. http://dx.doi.org/10.1016/j.cam.2022.114235.

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32

Chen, Tianran, and Tien-Yien Li. "Homotopy continuation method for solving systems of nonlinear and polynomial equations." Communications in Information and Systems 15, no. 2 (2015): 119–307. http://dx.doi.org/10.4310/cis.2015.v15.n2.a1.

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33

Romeuf, Jean-François. "A polynomial algorithm for solving systems of two linear diophantine equations." Theoretical Computer Science 74, no. 3 (1990): 329–40. http://dx.doi.org/10.1016/0304-3975(90)90082-s.

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34

Hakk, K. "TWO-GRID ITERATION METHOD FOR WEAKLY SINGULAR INTEGRAL EQUATIONS." Mathematical Modelling and Analysis 5, no. 1 (2000): 76–85. http://dx.doi.org/10.3846/13926292.2000.9637130.

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For the solution of weakly singular integral equations by the piecewise polynomial collocation method it is necessary to solve large linear systems. In the present paper a two‐grid iteration method for solving such systems is constructed and the convergence of this method is investigated.

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35

Rybalov, A. N. "On the generic complexity of solving equations over natural numbers with addition." Prikladnaya Diskretnaya Matematika, no. 64 (2024): 72–78. http://dx.doi.org/10.17223/20710410/64/6.

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We study the general complexity of the problem of determining the solvability of equations systems over natural numbers with the addition. The NP-completeness of this problem is proved. A polynomial generic algorithm for solving this problem is proposed. It is proved that if P ̸= NP and P = BPP, then for the problem of checking the solvability of systems of equations over natural numbers with zero there is no strongly generic polynomial algorithm. For a strongly generic polynomial algorithm, there is no efficient method for random generation of inputs on which the algorithm cannot solve the pr

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36

Bataineh, Ahmad Sami, Osman Rasit Isik, Moa’ath Oqielat, and Ishak Hashim. "An Enhanced Adaptive Bernstein Collocation Method for Solving Systems of ODEs." Mathematics 9, no. 4 (2021): 425. http://dx.doi.org/10.3390/math9040425.

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In this paper, we introduce two new methods to solve systems of ordinary differential equations. The first method is constituted of the generalized Bernstein functions, which are obtained by Bernstein polynomials, and operational matrix of differentiation with collocation method. The second method depends on tau method, the generalized Bernstein functions and operational matrix of differentiation. These methods produce a series which is obtained by non-polynomial functions set. We give the standard Bernstein polynomials to explain the generalizations for both methods. By applying the residual

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37

Vorontsov, Oleg, Valeriy Usenko, and Iryna Vorontsova. "DETERMINATION OF SUPERPOSITION COEFFICIENTS FOR DISCRETE FORMATION OF POLYNOMIAL FUNCTIONAL DEPENDENCIES." APPLIED GEOMETRY AND ENGINEERING GRAPHICS, no. 107 (February 26, 2025): 42–53. https://doi.org/10.32347/0131-579x.2024.107.42-53.

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The shape control of a discretely represented curve (DRC) in the static-geometric method can be achieved not only by varying the functional external load but also through the coefficients in computational templates. These templates form the basis for constructing systems of finite-difference equations for DRC formation and indicate the proportional contribution of adjacent nodes to the desired formation. This article proposes a general approach to creating computational templates for modeling geometric objects (GOs) using superpositions of point sets. This aims to further study the influence o

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38

Nigay, Ruslan, Evgeny Nigay, and Lubov Mironova. "Investigation of rigid dynamic systems on the example of modelling a tape drive mechanism." MATEC Web of Conferences 329 (2020): 03003. http://dx.doi.org/10.1051/matecconf/202032903003.

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The features of modeling the dynamics of mechanical systems on the example of the operation of the tape drive mechanism related to real technological processes are stated. An approach to solving stiff systems of differential equations by the numerical-analytical method is noted. The approach is based on solving systems of higher-order differential equations using elementary functions using procedures for precision search for the roots of the characteristic polynomial of the system. A mathematical model of the tape drive mechanism of a VCR is given as an example of a precision electromechanical

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39

Ajileye Ganiyu, Richard Taparki, Ojo Olamiposi Aduroja, and Rahimat Oziohu Onsachi. "Volterra Integral Equations: A Numerical Solution Method Using Shifted Chebyshev Polynomial." International Journal of Latest Technology in Engineering Management & Applied Science 14, no. 4 (2025): 940–44. https://doi.org/10.51583/ijltemas.2025.140400114.

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Abstract: This study presents a numerical method for solving Volterra integral equations of the second kind using shifted Chebyshev polynomials. Volterra integral equations arise in various scientific and engineering applications, including population dynamics, physics, and control systems. Due to their complexity, obtaining analytical solutions is often challenging, making numerical techniques crucial. We employ shifted Chebyshev polynomials as basis functions to approximate the solution, transforming the integral equation into a system of algebraic equations. The shifted Chebyshev polynomial

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40

Kashpur, O. F. "Conditions for the solvability of nonlinear equations systems in Euclidean spaces." Bulletin of Taras Shevchenko National University of Kyiv. Series: Physics and Mathematics, no. 1 (2021): 74–80. http://dx.doi.org/10.17721/1812-5409.2021/1.9.

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The solution of many applied problems is to ﬁnd a solution of nonlinear equations systems in ﬁnite- dimensional Euclidean spaces. The problem of ﬁnding the solution of a nonlinear system is divided into two problems: 1. The existence of a solution of a nonlinear equations system; in the case of nonunique of the solution, it is necessary to ﬁnd the number of these solutions and their surroundings. 2. Finding the solution of a system of nonlinear equations with a given accuracy. Many publications are devoted to solving problem 2, namely the construction of iterative methods, their convergence an

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41

Salim, S. H., R. K. Saeed, and K. H. F. Jwamer. "Solving Volterra-Fredholm integral equations by non-polynomial spline function based on weighted residual methods." Bulletin of the Karaganda University-Mathematics 117, no. 1 (2025): 155–69. https://doi.org/10.31489/2025m1/155-169.

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In this paper, a method that utilizes a non-polynomial spline function based on the weighted residual technique to approximate solutions for linear Volterra-Fredholm integral equations is presented. The approach begins with the selection of a series of knots along the integration interval. We then create a set of basis functions, defined as non-polynomial spline functions, between each pair of adjacent knots. The unknown function is expressed as a linear combination of these basis functions to approximate the solution of integral equations. The coefficients of the spline function are calculate

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42

GOULIANAS, K., A. MARGARIS, I. REFANIDIS, and K. DIAMANTARAS. "Solving polynomial systems using a fast adaptive back propagation-type neural network algorithm." European Journal of Applied Mathematics 29, no. 2 (2017): 301–37. http://dx.doi.org/10.1017/s0956792517000146.

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This paper proposes a neural network architecture for solving systems of non-linear equations. A back propagation algorithm is applied to solve the problem, using an adaptive learning rate procedure, based on the minimization of the mean squared error function defined by the system, as well as the network activation function, which can be linear or non-linear. The results obtained are compared with some of the standard global optimization techniques that are used for solving non-linear equations systems. The method was tested with some well-known and difficult applications (such as Gauss–Legen

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43

Sommese, Andrew J., Jan Verschelde, and Charles W. Wampler. "Advances in Polynomial Continuation for Solving Problems in Kinematics." Journal of Mechanical Design 126, no. 2 (2004): 262–68. http://dx.doi.org/10.1115/1.1649965.

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For many mechanical systems, including nearly all robotic manipulators, the set of possible configurations that the links may assume can be described by a system of polynomial equations. Thus, solving such systems is central to many problems in analyzing the motion of a mechanism or in designing a mechanism to achieve a desired motion. This paper describes techniques, based on polynomial continuation, for numerically solving such systems. Whereas in the past, these techniques were focused on finding isolated roots, we now address the treatment of systems having higher-dimensional solution sets

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44

Chistov, A. L. "Systems with Parameters, or Efficiently Solving Systems of Polynomial Equations: 33 Years Later. III." Journal of Mathematical Sciences 247, no. 5 (2020): 738–57. http://dx.doi.org/10.1007/s10958-020-04836-8.

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45

Chistov, A. L. "Systems with Parameters, or Efficiently Solving Systems of Polynomial Equations: 33 Years Later. I." Journal of Mathematical Sciences 232, no. 2 (2018): 177–203. http://dx.doi.org/10.1007/s10958-018-3868-z.

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46

Chistov, A. L. "Systems with Parameters, or Efficiently Solving Systems of Polynomial Equations 33 Years Later. II." Journal of Mathematical Sciences 240, no. 5 (2019): 594–616. http://dx.doi.org/10.1007/s10958-019-04378-8.

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47

Le, Huu Phuoc, and Mohab Safey El Din. "Solving parametric systems of polynomial equations over the reals through Hermite matrices." Journal of Symbolic Computation 112 (September 2022): 25–61. http://dx.doi.org/10.1016/j.jsc.2021.12.002.

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Ayad, Ali, Ali Fares, and Youssef Ayyad. "An algorithm for solving zero-dimensional parametric systems of polynomial homogeneous equations." Journal of Nonlinear Sciences and Applications 05, no. 06 (2012): 426–38. http://dx.doi.org/10.22436/jnsa.005.06.03.

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Wolf, T. "On solving large systems of polynomial equations appearing in discrete differential geometry." Programming and Computer Software 34, no. 2 (2008): 75–83. http://dx.doi.org/10.1134/s0361768808020047.

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Buchanan, S. Alasdair. "Some theoretical problems when solving systems of polynomial equations using Gröbner bases." ACM SIGSAM Bulletin 25, no. 2 (1991): 24–27. http://dx.doi.org/10.1145/122520.122523.

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